the point of isotropy and other properties of serial and parallel manipulators

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The “point of isotropy” and other properties of serial and parallel manipulators

G. Legnani a,⁎, D. Tosi a, I. Fassi b, H. Giberti c, S. Cinquemani c

a Università degli Studi di Brescia, Dipartimento di Ingegneria Meccanica e Industriale, via Branze 38, 25123, Brescia, Italyb Istituto Tecnologie Industriali ed Automazione ITIA — CNR, Via Bassini 15, 20133 Milano, Italyc Politecnico di Milano, Mechanical Department, Campus Bovisa Sud, via La Masa 34, 20156, Milano, Italy

a r t i c l e i n f o a b s t r a c t

Article history:

Received 20 March 2009

Received in revised form 6 May 2010

Accepted 12 May 2010

Available online 30 June 2010

As wellknown, the kinetostatic performances(repeatability, stiffness, maximum forceor velocity)

of serial and parallel manipulators depend strongly on the kinematic structure and on the

manipulator configuration inside its working space. The manipulator performances are often

analyzed using the manipulability ellipsoids which depend on the manipulator Jacobian.

This paper investigates the significance of the classical definition of manipulability ellipsoid

highlightingits lack of significance in somecircumstances andproposesa newextended definition

which takes intoaccountthe differentperformancesof each actuator.This definitionis particularly

useful for manipulators with different kinematic chains and various types of actuated joints (e.g.

revolute and prismatic).

Then the paper reviews the concept of isotropy and its properties: a manipulator exhibits an

isotropic behaviour when it has the same performances along all the directions of the working-

space. Then, the authors introduce the new concept of Point of Isotropy both for serial and parallel

manipulators, showing how in some circumstances a non-isotropic manipulator may be

transformed into an isotropic one simply changing the location of its TCP (Tool Center Point).This concept may be used to design new manipulators or to make isotropic already existing

manipulators just modifying the shape or dimension of the last link.

The theoretical investigation of this newconcept is supported by its application to the design of

an isotropic Stewart–Gough platform.

© 2010 Elsevier Ltd. All rights reserved.

Keywords:

Isotropy

Point of isotropy

Force isotropy

Velocity isotropy

Mass isotropy

Stiffness isotropy

Repeatability isotropy

PKM

Serial manipulators

Parallel manipulators

1. Introduction

The kinetostatic properties of a serial or parallel manipulator in term of achievable velocity, force, stiffness and motion

precision can be studied using the well known relations

Q = J S F s = J T

F q J =∂Q

∂S ð1Þ

where J is the Jacobian matrix which relates the gripper velocity S with those of the actuators Q , as well as the forces (or torques) F qexerted by them with the forces and the torques F s applied to the gripper

S = v x; v y; v z;ω x;ω y;ω z

h iT = V T

;ωT

h iT Q = q1; q2; q3; q4; q5; q6

Â ÃT

F s = f x; f y; f z; t x; t y; t z

h i= F T

; T T h i

T F q = f 1; f 2; f 3; f 4; f 5; f 6½ T

dS = d x; d y; d z; dα; dβ; dγ½ T = dX T ; dΦT

h iT

dQ = dq1; dq2; dq3; dq4; dq5; dq6½ T

ð2Þ

Mechanism and Machine Theory 45 (2010) 1407–1423

⁎ Corresponding author. Tel.: +39 030 3715 425; fax: +39 030 3702 448.

E-mail address: [email protected] (G. Legnani).

0094-114X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2010.05.007

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where the elements of Ṡ are composed by the linear velocity V of the TCP (Tool Center Point) and the angular velocity ω of the end-

effector, qi is the ith actuator coordinate and q i the corresponding velocity, f i is the ith actuator force (or torque), and the elements

of F s are the forces exerted by the TCP and the torques exerted by the end-effector. Finally d X and dΦ are the infinitesimal TCP

displacements and rotations while dQ = J dS are infinitesimal motions of the actuators. Since J appears both in the velocity and in

the force/torque relations, the set of the two Eq. (1) is said to represent the “kinetostatic duality”.

Generally the velocity, the force and the compliance that a manipulator can exert depend on the direction along which they are

performed as well as on the manipulator configuration. The manipulator may have singular configurations in which (or near

which) the performances in some directions are extremely poor while in others are extremely good. Conversely, the manipulator

may have isotropic configurations where the performances are identical along all directions.

Isotropy is a key issue in improving serial and parallel manipulators performances. Much research has been carried on to design

fully isotropic manipulators. Some results have been obtained for the case of 3-DoF (degree of freedom) manipulators [1–3] and

two different procedures for the design of 6-DoF PKM (parallel kinematic machines) have been presented in [4,5].

The concept of isotropy is widely discussed in [6] with reference to several dexterity indices developed for serial and parallel

robots.

Performances, isotropy and singularity are often studied on the bases of the eigenvectors and of the eigenvalues of some matrix

related to the Jacobian. Other important parameters to be investigated are the determinant and the condition1 number of J .

Note 1. As mentioned in Eq. (1), in this paperwe adopt as definition of Jacobian J =∂Q /∂S which is that usually adopted for parallel

manipulators PKM, while for serial manipulators the Jacobian is usually defined as J =∂S /∂Q.

2. The classical approach to isotropy

This section critically discusses and reviews the main available results on isotropy.

2.1. General de finitions

The concept of robot isotropy was firstly introduced in 1982 by Salisbury and Craig, for the design of manipulator hands with

serial kinematics [7]. Since then, much research has been carried out for serial and parallel manipulators, especially for kinetostatic

design and control purposes.

A robot is called “isotropic” if at least in one point of the working space some of its kinetostatic properties are homogeneous

with respect to all the directions. These properties can be investigated by means of the manipulability ellipsoids [8].

The main thrust on isotropy and isotropic manipulators was given by Angeles and his co-workers. Angeles and López Cajún [9]

defined a serial non-redundant manipulator as isotropic when its Jacobian matrix satisfies the relation

J T J = λI 6Â6 ð3Þwhere λ is a scalar and I 6×6 is the 6× 6 identity matrix. However to give significance to this condition it is essential to normalizethe

units by dividing all the manipulator lengths by a proper scalingfactor (“characteristic length”) [10]. Unfortunately it is not always

clear how this length must be selected. The choice is task dependent. A convenient methodology to automatically evaluate the

characteristic length is presented in [4]. Alternative solutions may be obtained by the study of 3D rigid body motion [11].

The use of Eq. (3) is not always satisfactory and sometimes to take into account the characteristics of the manipulator or of the

actuators, the matrix J T J is replaced by H (or its inverse)

P

H = J T

HJ ð4Þwhere H = H T is a proper weighting matrix to be suitable defined, depending on the characteristic to be highlighted. By means of

the kinetostatic duality, in many situations it is possible to achieve simultaneously isotropy in velocity, force and stiffness (for

instance when H is the identity matrix). Isotropy implies that the condition number of J is 1, while in singular confi

guration it isinfinitive and the determinant is null (or infinity).

In particular, considering a 3-DoF spatial manipulator with pure translational motion, and thus S = [ x, y, z]T , the following

definitions can be applied

• Velocity isotropy. A manipulator is isotropic with respect to the velocity, if it can perform the same velocity along all the

directions. This is investigated by means of the velocity ellipsoid.

• Force isotropy. A manipulator is isotropic with respect to the force, if it can exert the same force along all the directions. This

property can be investigated by means of the force ellipsoid.

• Stiffness isotropy. A manipulator is isotropic with respect to the stiffness, if the deflection of the TCP produced by a force applied

to it is always in the direction of the force and its magnitude is independent of the force direction. This is analyzed by means of

the stiffness ellipsoid.

1

For a general matrix A the condition number is defined as cond( A) =∥ A∥ ∙ ∥ A−1

∥. If the “spectral norm” is considered (the maximum singular value of A), thecondition number equals the ratio between the maximum and the minimum of the singular values cond( A) =σ max( A) /σ min( A).

1408 G. Legnani et al. / Mechanism and Machine Theory 45 (2010) 1407 –1423

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• Repeatability isotropy. A manipulator is isotropic with respect to the repeatability, if the region of uncertainty of the TCP

position, due to uncertainty in the actuators motion, is a sphere. This is analyzed by means of the repeatability ellipsoid.

• Mass isotropy. A manipulator is isotropic with respect to the equivalent grippermass, if the acceleration of the TCP produced by a

force applied to it is always in the direction of the force and its magnitude is independent of the force direction. This is analyzed

by means of the mass ellipsoid.

These concepts can be easily extended to general 6-DoF manipulators.

As highlighted in [6,12] and visualized in Figs. 1–10, some kinetostatic properties related to velocity, force, and/or displacement

are better represented by parallelograms (or polyhedra) rather than ellipses (or ellipsoids), but ellipses are generally adopted

because they can be evaluated more easily and give approximated but useful information (Fig. 1). For stiffness and mass, ellipses

constitute the correct representation. Parallelograms and ellipses are used for planar 2D cases, polyhedra and ellipsoids for spatial

3D studies.

Figs. 1–10 illustrate some concepts discussed in the paper. Parallelograms are sometimes sketched with curved edges to

suggest a possible geometrical constructions, but the correct definition requires straight segments. For simplicity the pictures refer

to 2D planar manipulators, but the paper discussion concerns 6-DoF manipulators.

2.2. Velocity isotropy

The velocity ellipsoid is generally defined imaging to move the manipulator constraining the joint velocity as

Q T Q = 1 ð5Þ

which combined to Eq. (1) leads to

S T J

T J S = 1 ð6Þ

This relation defines an ellipsoid in the TCP velocity which describes the capacity of the manipulator to generate velocity in the

different directions. The eigenvalues of J T J are the reciprocal of the square of the semi-diameters of the ellipsoid and the

corresponding eigenvectors represent the directions of the ellipsoid diameters [8]. These diameters are orientated along the

directions in which the TCP velocity is maximum or minimum.A 6-DoF spatial manipulator is isotropic with respect to velocity if the matrix J T J is diagonal

J T J = diag j xx; j yy; j zz; jαα ; j ββ ; jγγ

with

j xx = j yy = j zz

and

jαα = j ββ = jγγ

8>><>>: ð7Þ

In this case the ellipsoids for linear and angular velocities are spheres (see Fig. 3).

The definition of isotropy given by Eq. (7) is similar to that given by Eq. (3) but it has the advantage that does not require the

normalization of the Jacobian by means of the characteristic length. Moreover Eq. (7) treats separately the translational and the

rotational behavior of the gripper. Indeed this formulation is not still adequate if the actuators are different to each other (seeSection 3.1).

Fig. 1. Example of ellipses and parallelograms representing maximum force or velocity that can be generated by a 2-DOF manipulator; left: isotropic situation,

right: general case.

1409G. Legnani et al. / Mechanism and Machine Theory 45 (2010) 1407 –1423



Fig. 3. The isotropic configuration of a SCARA robot which requires: l1 = ffiffiffi

2p

l2 , β =±3/4π .

Fig. 4. The velocity ellipsoid for a polar manipulator.

Fig. 2. Construction of the repeatability ellipse for a SCARA robot. In the singular configuration (dashed) the ellipse collapses to a segment. With the classical

definition, repeatability and velocity ellipses coincide.

Fig. 5. The force ellipses for a polar robot; f and t are the force (torque) of the actuators.




2.3. Force isotropy

The definition of the force ellipsoid is similar to the velocity one

F T q F q = F

T S J

−1 J −T

F s = 1 ⇒ F

T S J

T J

−1

F S = 1 ð8Þ

So the force ellipsoid is associated to the eigenvectors and eigenvalues of ( J T J )−1. By comparing Eqs. (6) and (8) we conclude

that the force and the velocity ellipsoids are reciprocal to each other since the eigenvalues of any matrix A−1 coincide with the

inverse of those of A while the eigenvectors are the same. Thus the directions of maximum available velocity coincide with those of

the minimum available force and vice versa (compare Figs. 4 and 5). The volumes of the force and velocity ellipsoids are also

reciprocal to each other.

2.4. Repeatability

The repeatability of the manipulator TCP due to small indetermination in the actuator motions is also represented by the

velocity ellipses. In fact the equation describing infinitesimal motion is similar

dQ

dt = J

dS

dt ⇒ dQ = JdS ⇒ dS

T J

T JdS = 1 ð9Þ

This ellipse is a simplified representation of the uncertainty position area of the gripper due to inaccuracy of the actuator motion.

Fig. 6. Educational example of mass ellipse: in the Cartesian manipulator the equivalent mass in x direction is m1+ m2, and in y direction is m2.

Fig. 7. Construction of the mass ellipse for a 2-DOF planar manipulator.




2.5. Stiffness isotropy

Assuming that the actuators are locked and that they are the only sources of compliance, the force F s to be applied to the end-

effector to produce a motion dS is

F s = J T

K q JdS K q = diag …; ki;…ð Þ ð10Þ

where ki is the stiffness of the ith actuator. This equation is easily obtained from Eq. (1), and remembering that

F q = K qdQ with f i = kidqi

The manipulator compliance is synthetically represented by its stiffness matrix K s

K s = J T K q J ð11Þ

A general 6-DoF manipulator is fully isotropic with respect to stiffness if K s has the following diagonal structure

K s = diag k xx; k yy; k zz; kαα ; k ββ ; kγγ

with

k xx = k yy = k zz = k x

kαα = k ββ = kγγ = kϕ

(ð12Þ

Fig. 9. Velocity ellipses for a Cartesian manipulator (extended definition) when the velocity of the first joint is higher than that of the second.

Fig. 8. Velocity ellipses for a Cartesian manipulator (classical definition).




In this case it results

F = k xdX

T = kϕdΦ

(ð13Þ

where k x is translation stiffness and kϕ is rotation stiffness. This means that

• forces F applied to the TCP do not produce rotations dϕ but only translations d X ;

• the translation is proportional to the force and parallel to it regardless to the force direction;

• torques T applied to the TCP do not produce translations d X but only rotations dϕ;

• the rotation is proportional to the torque and occurs around the same axis as the applied torque.

In accordance to Eq. (3), the manipulator is isotropic with respect to stiffness if it exists a scalar λ for which

cond K LK sK Lð Þ = 1 K L = diag 1; 1; 1; 1 = λ; 1 = λ; 1 = λð Þ ð14Þ

If the manipulator is isotropic, K s has the structure of Eq. (12) and soλ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kϕ = k x

p . This conditionis very similar to that adopted

to normalise the Jacobian using the characteristic length [4,13].

In the special case in which all the actuators are identical to each other and therefore they have the same stiffness k, Eq. (11)

simplifiesas K s= kJ T J and the condition number of the matrix J T J can be investigated instead of that of J T K q J . In this case the isotropy

for velocity, force and stiffness are achieved contemporarily when the condition of Eq. (3) is satisfied.

With reference to Eq. (12), a manipulator is partially isotropic if it yields

k xx = k yy≠k zz and= or kαα = k ββ ≠kγγ ð15Þ

The concept of partial isotropy can be exploited in the design of manipulators when a different stiffness may be required in a

particular direction. For example when a 6-axis manipulator is employed as milling machine (e.g. Celerius manipulator [14]) the

stiffness along the z direction of the TCP (spindle axis) may be higher than that in the xy plane; similar criterion may be applied to

torque. This concept of partial isotropy may be extended also to velocity, force, and mass.

A graphical representation of the stiffness ellipse can be derived from that of the mass (Fig. 7) simply substituting dS for S and

stiffness constants for masses.

2.6. Mass isotropy

It is well known that the actuator forces (or torques) necessary to actuate a manipulator can be expressed as

F q = MQ ::

+ V Q ;Q

Where M = M (Q ) is a suitable symmetrical square mass matrix (positive defined) and V is a suitable vector depending on the

masses and on the movement. Remembering the relations F s= J T F q and Q ≃HS (for small velocity) it is possible to relate a force

applied to the gripper with its acceleration as

F s

= J T

MJS ::⇒ F

s

=―M S

::

ð16

Þwhere an ellipse can be associated to the matrix M = J T MJ to represent the force needed to produce a unitary acceleration of the

gripper (Figs. 6 and 7). In case of isotropy a gripper force produces a pure translational acceleration and a torque produces a pure

angular acceleration of the gripper. The acceleration of the gripper is parallel to the force and its magnitude is independent of its

direction. The same happens for torque and angular acceleration. This happen when

―M = diag m; m; m; J g ; J g ; J g

where m is an equivalent mass and J g is an equivalent inertia moment.

3. An extended approach to isotropy

This section highlights the limits of the classical definitions of isotropy and proposes a new extended one.




3.1. Normalisation of Jacobian

From thediscussion of theprevioussectionsit is evident that in thecasesconsidered by theclassical definitions, the isotropy for force,

velocity and stiffness coincide to each other. Summarizing the results, the kinetostatic properties of a manipulator are studied on the

bases of the properties of the matrix J T J or its inverse. The definition of the manipulability ellipsoids and of isotropy are based on them.

However, the classical definition of isotropy holds under the hypothesis that all the actuators are identical, or at least of the

same type (revolute or prismatic) but it is meaningless when the actuated joint are different; in fact in these case the elements of

the Jacobian have different units. For example in the case of the polar robot (Figs. 4 and 5) we get

S = x

y

" #Q =

α

ρ

" #J =

∂Q

∂S =

∂α

∂ x

∂α

∂ y

∂ρ

∂ x

∂ρ

∂ y

266664

377775

( x = ρcos α

y = ρsin α

(α = arctan

y

x

+ kπ

ρ = xcos α + ysin α

ð17Þ

and so the units U of matrices J and of J T J are

U J ð Þ =1ℓ

1ℓ

1 1

24 35 U J T J

=

1 +

1

ℓ2 1 +

1

ℓ2

1 +1

ℓ2

1 +1

ℓ2

26643775 ð18Þ

where ℓ stands for length and 1 for “pure numbers”; the terms are inconsistent in units. It is evident that in this case expressing

the units, for example, in meters or millimeters would change the values of the elements of J modifying the shape of the ellipses.

This observation highlights the loss of significance of the classical definition for isotropy. In fact the isotropy, and so the ellipses

shape, must be a property of the manipulator regardless the choice of the adopted units. To overcome this problem a further

characteristic length may be adopted to scale the manipulator dimensions [15]. However the choice of the value of the

characteristic length is arbitrary, and there is not a “universal” choice which is suitable for all the situations. Various approaches

may be proposed depending on the characteristics to be highlighted.

In this paper we propose a choice for the scaling factors that considers the nature of the actuators, and it is based on the

maximum achievable performances of each actuator. For the velocity ellipsoid we propose to use its maximum achievable velocity

q max,i and for force ellipsoid its maximum performable force (or torque) f q,max,i. This choice gives a physical and concrete value tothe scaling factors. The actuator forces and velocities are then scaled as

―Q =

Àq1

⋮Àqi

⋮

2666664

3777775 =

1 =qmax;1 0

⋱

1 =qmax;i

0 ⋱

2666664

3777775

q1

⋮

qi

⋮

2666664

3777775 = K

−1v Q

Àqi =

qi

qmax;i

≤ 1 ð19Þ

―F

q=

― f q;1

⋮

― f q;i

⋮

2

66666664

3

77777775=

1 = f max;1 0

⋱

1 = f max;i

0 ⋱

2

66666664

3

77777775

f q;1

⋮

f q;i

⋮

2

66666664

3

77777775= K

−1

f F

― f

q;i=

f q;i

f max;i

≤ 1ð20

Þ

The definitions for the force and velocity ellipses are consequently changed to

ÀQ

T ÀQ = 1

ÀF q

T ÀF q = 1 ð21Þ

and, since K v= K vT and K f = K f

T , we have

S T

J T

K −2v J

S = 1 F

T s J

T K

2 f J

−1

F s = 1 ð22Þ

Eqs. (21) and (22) express the extended definition for velocity and force velocity.

Using these defi

nitions the ellipses express the kinetostatic performances of the manipulators depending on the adoptedactuators which have specific maximum velocity and force (or torque).




For example in the case of Cartesian manipulator (Figs. 8 and 9) if the maximum velocity of the first link is higher than that of

the second one, the diameters of the ellipses change accordingly.

As a further example, when analyzing the repeatability, the manipulability ellipsoid and the isotropy condition may be defined

as

dS T J

T K

2r JdS = 1 cond J

T K

2r J

= 1 K r = diag

1

δq1

;…;1

δqi

;…

ð23Þ

where δqi is the measures of the uncertainty in the motion of the ith actuator.

3.2. Generalized isotropy

Summarizing the results, a manipulator is said to be in an isotropic configuration if the matrix J T HJ has the following diagonal

structure

J T

HJ = diag α;α;α;β;β;βð Þ ð24Þ

where H is a suitable weighting matrix depending on the property under investigation, and α , β are scalar. It is important to note

that using the extended definition for ellipses and isotropy, force isotropy and velocity isotropy are achieved simultaneously only if

it exists a scalar k for which

K v = kK −1 f ð25Þ

Similar deductions can be made for the simultaneous achievement of stiffness and velocity or force isotropy.

If the weights of all the actuators assume an identical value, the proposed extended definition for the manipulability ellipsoids

and for isotropy coincide with the classical ones.

4. The point of isotropy for a general 6-DoF manipulator

In this section we propose a methodology to discover if, for a given manipulator configuration, there is one or more choices of

the TCP location for which the manipulator is isotropic. In other words, a serial or parallel manipulator has a “point of isotropy” ifit

exists at least one point of its end effector for which the isotropy condition is achieved. This concept may be used to design new

manipulators or to make isotropic already existing manipulators just modifying the shape or dimension of the last link.

According to our knowledge this concept has never be studied since now.

The manipulability ellipses for force, velocity and stiffness are related to the manipulator Jacobian which in turn dependson the

manipulator geometry (link lengths, joint type and location,…) as well as on the position of the TCP. For this reason if the location

of the TCP is changed, the ellipses also change (Figs. 10 and 11).

Conditions expressed by Eqs. (7), (8), or (22) determine if the manipulator, with the chosen TCP, is isotropic in the analyzed

configuration.

We assume that J is the manipulator Jacobian and dQ = J dS the infinitesimal motion of the actuators that produces a TCP

displacement dS . We consider now another point TCP′ embedded on the end-effector which is located at a distance P = [ p x p y p z ]T

from the TCP. Its motion dS ′ will be dS ′= J dS

dS ′ =dX ′

dΦ

!= I − X

P

0 I

!dX dΦ

!=

À JdS

À J −1

= I X P

0 I !

X P =

0 − p z p y

p z 0 − p x

− p y p x 0

24 35ð26Þ

where d X stands for translations and dΦ stands for rotations; d X ′= d X +dΦ× P = d X −P × dΦ.

The new Jacobian J ′ will be evaluated in this way

dQ = JdS = J ′dS

′with ⇒ J

′= J

P

J −1 ð27Þ

in this case the manipulability ellipses will be defined as

S ′T

J ′TH J

′S ′

= 1⇒ S ′T P

J T J

T HJ

P

J

S ′

= 1 ð28Þ

F ′

qT HF ′

q = 1⇒F ′

sT P

J T J T HJ P

J −1

F ′

s = 1 ð29Þ




and, noting that P _ =−P _T , the condition for isotropy will be

ð30Þ

where α and β are scalars, I is a 3×3 identity matrix and

ð31Þ

Eq. (30) can be expanded to

ð32Þ

and it is easy to verify that the isotropy conditions of Eq. (32) are achieved only if the following conditions are satisfied

J ff = αI J ft = − J T

ft skew−symmetricð Þ J ′tt = J tt +1

α J

2 ft = βI ð33Þ

the corresponding location of the TCP′ with respect to the TCP will be

―P = −

1

α J ft ð34Þ

Fig. 11. The kinetostatic parameters of two points TCP and TCP′ embedded on the gripper at a distance P are related to each other.

Fig. 10. Velocity ellipses for a SCARA robot. If the gripper location is modified from TCP to TCP′, the ellipses changes accordingly achieving isotropy.




Summarizing, the isotropy is possible only in the TCP location expressed by Eq. (34). More in detail, the search for an isotropic

TCP′ can be performed with the following steps

if J ff ≠α I then

TCP′ does not exist

else if J ft ≠− J ft T then

TCP′ does not exist

else if J tt = J tt + 1α J 2 ft ≠βI thenTCP′ does not exist

else

TCP′ exists in position―P = −

1

α

J ft

end

Note 2. An important consequence of Eq. (33) is that considering a specific configuration of any 6-DoF serial or parallel

manipulator, it may have at most one possible choice for the TCP that makes it isotropic. Under the conditions of Eq. (7),

degenerated cases may happen only for α = j xx= j yy= j zz =0 (i.e. J ff =0). In this case any values for P satisfy the equations.

However in this case the manipulator is in a singular configuration which is generally to be avoided.

Note 3. The details of the discussion of this section concern the velocity isotropy or the repeatability one, but since all the

considered isotropy are based on ( J T HJ )h with a suitable weighting matrix H and h =1 or h =−1, the results are immediately

generalized to all the other isotropy.

5. Application to Stewart–Gough platform

In this section the theory about the point of isotropy is applied to the design of a parallel manipulator.

5.1. The Jacobian

Fig. 12 shows a schematic representation of a Stewart–Gough platform; it is a 6-DoF PKM actuated by extensible legs. Its

Jacobian assumes the following structure [4,5]

ð35Þ

where wi is the unit vector of the ith leg and pi is the vector giving the position of the ideal centre of the ith spherical joint with

respect to the TCP; r i is the vector defining the minimum distance of the line defined bythe ithleg fromthe TCP, ui is the moment of

the unit vector of the leg wi with respect to the TCP. The position of the spherical joints of the fixed base is pi#. The Jacobian depends

only on the leg directions and on their distance from the TCP, the actual joint positions along the line are not important ( Fig. 13).

In the following sections two application cases are analyzed. The first manipulator considered does not have the point of isotropy. While the second, with a proper choice of the TCP location, exhibits an isotropic behavior.

5.2. A non-isotropic hexapod

Usually, the isotropic behavior is associated with geometrical symmetry in the manipulator design. Hence, the architecture

shown in Figs. 14–17 seems a good candidate.

It is assumed that the TCP is in the center of the mobile base, with z axis directed upward and that the six legs are “uniformly”

distribute as shown in Fig. 14. The joints on the fixed and the mobile base are located on two circles. The distance between the

fixed and the mobile bases is arbitrary. The legs are divided into two groups; group 1 includes legs 1, 2, and 3 and group 2 collects

legs 4, 5, and 6. As proved in [4] the elements of J T J are not affected by a rigid rotation of any of the groups of the legs around z axis

(Fig. 15); θ is the angle that each leg forms with the z axis, with r is the minimum distance of each leg from the z axis, and h is the

vertical distance of the TCP with respect to the plane η defi

ned by the segments of minimal distance between the legs and the z axis(Figs. 16 and 17).




Once the size of the fixed and of the mobile platform has been fixed, the angle θ depends on the legs length and so on the vertical

displacement of the mobile base.

With this assumption, we get

ð36Þ

where for H = I [4]

J ft =

0 3hsin2θð Þ 0

−3hsin2

θð Þ 0 0

0 0 0

266664

377775 J ff = 3

sin 2θð Þ 0 0

0 sin2

θð Þ 0

0 0 2cos2θð Þ

266664

377775

J tt =

r 2−h2

cos2θð Þ + h2 0 0

0 r 2−h

2

cos2θð Þ + h

20

0 0 2r 2sin2θð Þ

266666664

377777775

Fig. 13. If the location of a spherical joint is moved on the line defined by the leg, the Jacobian does not change. Actuators may be located at either side of theplatform.

Fig. 12. A schematic representation of a Stewart–Gough platform and its reference systems.




Fig. 14. Hexapod with “regular” distribution of the legs (top view — xy plane). Legs 1, 2, and 3 solid lines, legs 4, 5, and 6 dashed.

Fig. 15. The matrix J T J does not change if one (or both) group(s) of legs are rotated by an arbitrary angle (top view — xy plane).

Fig. 16. The disposition oflegs 1,2, and 3;(legs4, 5,and6 are “symmetrical”). Plane η is the plane containing the segmentsof minimaldistance between the legsand z axis.




Fig. 18. The disposition of the six legs: 1, 2, and 3 continuous line; legs 4, 5, and 6 dashed line.

Fig. 17. The disposition of the unit vectors of the six legs: 1, 2, and 3 continuous line; legs 4, 5, and 6 dashed line.

Fig. 19. One possible disposition of the spherical joints and the TCP to achieve isotropy.




the condition on J ff requires

sin2θð Þ = 2cos

2θð Þ⇒ θ = arccos

ffiffiffi3

p

3

⇒ α = 2 ð37Þ

and a candidate for TCP′ is:

―P = −

1

2 J ft =

0 −h 0h 0 00 0 0

24

35⇒ P =

00h

24

35 ð38Þ

but

J ′ = J tt +1

α J

2 ft =

r 2 0 00 r 2 00 0 4r 2

24

35≠ βI ð39Þ

and so we conclude that the analysed PKM, in the considered poses, does not have any isotropy point.

5.3. A hexapod that can be made isotropic

Let us now consider a modification of the previous example where the planes containing the segment of minimum distance of

the two groups from axis z have different altitudes. We indicate them with h and k (Fig. 18).

The value of J ff is unchanged with respect to example1 because it depends just on the direction of the legs. The inclination angle

is then the same θ = arccos ffiffiffi

3p

= 3

and so α =2. Conversely we get

J ft =

03

2h + kð Þsin

2θð Þ 0

−3

2h + kð Þsin

2θð Þ 0 0

0 0 0

26666666664

37777777775

J tt = 3

Φ 0 0

0 Φ 0

0 0 2r 2

sin2θð Þ

264

375 with Φ = r 2−

h2 + k2

2

cos2

θð Þ +h2 + k2

2

ð40Þ

and for θ = arccos ffiffiffi

3p

= 3

a candidate for TCP′ is

―P = −

1

2 J ft =

0 −

h + k

2 0

h + k

20 0

0 0 0

26666643777775 ⇒ P =

00

h + k

2

2666437775 ð41Þ

and

J ′ = J tt +1

α J

2 ft =

r 2 +1

2h−kð Þ2

0 0

0 r 2 +1

2h−kð Þ2

0

0 0 4r 2

2666664

3777775

= βI ð42Þ




which is satisfied for

h−k = Æ ffiffiffi

6p

r ð43Þ

The two solutions for Eq. (43) produce two configurations symmetrical with respect to x or y axis. We conclude that, for each

configuration, the PKM has exactly one “point of isotropy” and that TCP′ must be located between planes η 1 and η 2 and equispaced

from them (Fig. 19)

P =

0

0P

z

264

375 =

0

0

h + k

2

266664

377775 =

0

0

h + λ

ffiffiffi6

p

2r

2666664

3777775 k = h + λ

ffiffiffi6

p r λ = Æ 1 ð44Þ

where λ=±1 indicates the two different configurations of Fig. 20. For each configuration just one TCP′ exists (Fig. 21).

In the considered example it has been chosen, for simplicity, that in the isotropic configuration all the six legs form the same

angle with respect to the z axis. However this is not strictly required; as shown in [4] the angle of the legs must satisfy the

following constrain

1−3cos2θ1ð Þ

1−3cos2 θ2ð Þ = −1 ⇒ θ2 = arccos1

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6−9cos θ1ð Þ

p !ð45Þ

where θi is the inclination angle of the ith group of legs. The distances r 1, r 2 of the two groups of legs from z axis and the distances

h′= h− z , k′= k− z of the TCP′ from planes η 1 and η 2 must be also adjusted accordingly to the following equations

r 1sin θ1ð Þcos θ1ð Þr 2sin θ2

ð Þcos θ2

ð Þ

= −1

h′sin

2θ1ð Þ

k′sin2θ2ð Þ

= −1

r 21 + h

′2−3r

21

sin

2θ1ð Þ

r 22 + k′2−3r 22À Á

sin2 θ2ð Þ = −1

ð46Þ

The minus signs in the right hand terms of Eq. (46) impliesthat the TCP is between planes η 1 and η 2, and thattwo groupsof legs

must lie on “opposite sides” of the circles of minimum distance (see Fig. 20).

Fig. 20. Top view of two alternative configurations of the isotropic PKM of Figs. 18 and 19.




Eqs. (45) and (46) form a set of four equations in the six variables θ1, θ2, r 1, r 2, h, k. Designing the isotropic manipulator we can

choose one of the angles according to Eq. (45) and one of the distances r 1 or r 2. So an infinitive number of isotropic PKM can be

obtained. For each choice of the parameters (i.e. for each PKM) a unique value of h and k is found, thus the location of TCP′ is

unique.

This example confirms that, at least for the PKM configurations centred with respect to the z axis and for the considered family

of Stewart–Gough platforms there is at most one position of the TCP with respect to the mobile base that makes them isotropic.

6. Conclusions

The paper presents different issues about kinetostatic performances of serial and parallel manipulators. First of all a new

definition of manipulability ellipsoid based on a weighted Jacobian is given. We propose to choose the weighting factors depending

on the actuators performances (i.e. maximum velocity, force, best accuracy,…). The main advantage of this new definition is that

the actuators characteristics are taken into account while estimating performances indices. The result is particularly useful for the

manipulators in which the actuators are different for typology (prismatic and revolute) or performance (different stiffness,

maximum velocity or force/torque).

In the second part of the paper isotropy is further investigated, introducing the new concept of point of isotropy of a generic

serial or parallel manipulator. It is shown how, in some cases, a manipulator becomes isotropic simply changing the location of its

TCP (i.e. the shape or the size of the end-effector). Each manipulator may have at most one point of isotropy. We believe that this is

a useful concept for the practical design of isotropic manipulators (serial or parallel). This new issue is completely discussed by a

theoretical point of view and then it is applied,as an example, to a Stewart–Gough platform. Relevant statements on the number of

possible isotropic configurations are presented.

References

[1] G. Hanrath, G. Stengele, Machine Tool for Triaxial Machining of Work Pieces, United States Patent No 6328510 B1 (Specht Xperimental 3.axis Hybrid MC),(2001).

[2] M. Schwaar, J.Kirchner, R. Neugebauer, PCT, WO 00/44526, (2000).[3] A. Pönish, Werkzeugmaschine mit Koppelfuhrung, Patentschrift, DE 19836624 C1 (SKM 400 3-axis parallel MC), (1998).[4] I. Fassi,G. Legnani, D. Tosi, Geometrical conditions forthe designof partial or full isotropichexapods, Journal of Robotic Systems 22(10) (2005)507–518, doi:

10.1002/rob.20074 .[5] K.Y. Tsai, K.D. Huang, The design of isotropic 6-DoF parallel manipulators using isotropy generators, Mechanism and Machine Theory 38 (2003) 1199–1214.[6] J.P. Merlet, Jacobian, manipulability, condition number, and accuracy of parallel robots, Journal of Mechanical Design 128 (1) (2006) 199–206.[7] J.K. Salisbury, J.J. Craig, Articulated hands: force control and kinematic issues, International Journal of Robotic Research 1 (1) (1982) 4–17.[8] T. Yoshikawa, Foundations of Robotics: Analysis and Control, MIT Press, Cambridge, MA, 1990.[9] J. Angeles, C.S. López-Cajún, Kinematic isotropy and the conditioning index of serial type robotic manipulators, International Journal of Robotics Research 11

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[11] J. Angeles, Is there a characteristic length of a rigid-body displacement? MMT Mechanisms and Machines Theory 41 (2006) 884–896, doi:10.1016/j.mechmachtheory.2006.03.010.

[12] S. Krut, O. Company, F. Pierrot, Velocity performance indexes for parallel mechanisms with actuation redundancy, Proceedings of the WORKSHOP onFundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators October 3.4, 2002, Quebec City, Quebec, Canada, 2002.

[13] J.M. Rico, J. Duffy, Robot isotropy: a reassessment, Proceedings of Sixth International Symposium on Advances in Robot Kinematics: Analysis and Control,Kluwer Academic Publishers, Strobl, Austria, June 1998.

[14] L. Molinari-Tosatti, I. Fassi, G. Legnani, Kinetostatic optimisation of PKMs, Proceedings of 53rd CIRP General Assembly, Montreal, 2003.[15] K.E. Zanganeh, J. Angeles, Kinematic isotropy and the optimum design of parallel manipulators, International Journal of Robotics Research 16 (2) (April 1997)

185–197.

Fig. 21. 3D view of the isotropic PKM of Fig. 20.


http://dx.doi.org/10.1002/rob.20074








the point of isotropy and other properties of serial and parallel manipulators

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